Philosophy babble: my personal inquiry to formal logic

Since my previous entry on ethics was better received than expected (i didn’t really anticipate any noticing), i now feel tempted to write another entry on the subject of philosophy. This time down the path of logic, but beware that this text will only be scraping on the surface of something deeper and more extensive than i have hence familiarized myself with.

Since reading a biography about Ludwig Wittgenstein some time ago, i felt compelled to look into the subject of formal logic, as approached by Gottlob Frege and Bertrand Russell, and then expanded upon by Wittgenstein. The pursuit of creating a perfect language of logic seemed a bit overambitious and i didn’t check the original work, particularly Tractatus Logico Philosophicus (further referenced as TLP), until recently.

This is a humble attempt however, to convey my interpretation and connect it with a few other ideas.
The main conclusion in TLP is that the relation between language and facts cannot itself be expressed in language. And that which cannot be thought of, cannot be expressed in language, and thereof one shall be silent.

TLP is a rather odd but eloquent book, structured into propositions numbered to clarify their order and with decimals indicating if a proposition follows from the previous one.

In the preface Wittgenstein seems to proceed from the premise that a limit can only exist if something can be known about both sides of it. He says ”The book will therefore draw a limit to thinking, or rather -not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think of both sides of this limit (we should therefore have to be able to think what cannot be thought)”. At least to me, this required some contemplation of the reader to understand why this premise is true (i was trying to think of a mathematical example to disprove it, but insufficient mathematical knowledge prevented me from doing so). In trying to understand something it is often useful to put it in your own words; If one cannot envision both sides of a limit, one cannot know if the limit exists. Hence this becomes an epistemological problem instead of a logical one. Although it remains unclear whether Wittgenstein makes this distinction. After all, he is talking about thoughts which are often constrained by epistemology, in the sense that our thoughts about the world is constrained by what we can know about the world. In a way, this insight would have been sufficient for Wittgenstein to arrive at his final conclusion, but there are other steps and parts of the book, perhaps not easily available to a general reader like me, but still of importance.

Let’s move on to the terms Wittgenstein uses in his propositions. Here, the introduction by Russell was of great help in making sense of them (although Wittgenstein himself claimed this introduction was riddled with misunderstandings).
A ”fact” is anything that can be true or false. Conversely, facts are what make propositions true or false. If a proposition (statement or sentence) is broken down to its smallest possible constituents, we get something called ”simples” or ”objects” (apparently synonymous, but i prefer ”object” and shall use it further). In turn, objects can be part of ”atomic facts”, which are parts of propositions, that if reduced further, cannot be true or false. Thus ”atomic facts” are indivisible, or at least lose their status as facts if divided. For example if ”Socrates is wise” is an atomic fact, then ”Socrates” and ”wise” are its objects.
From Wittgenstein’s propositions about these terms, and proposition 1.13 ”The facts in logical space are the world”, Russell deduces ”The world is fully described if all atomic facts are known, together with the fact that these are all of them”.
A proposition stating an atomic fact is conveniently named ”atomic proposition”. All atomic propositions are logically independent of each other, which means they don’t imply one another or follow from one another, and neither are they inconsistent.
Propositions used in logical inference however, are named ”molecular”. These can be inputs in truth functions by using logical operators, AND, OR and NOT. In conventional formal logic (for exceptions, see fuzzy logic) a truth function, f(x), can take on either of two values, true or false. The variable x is molecular, for example the proposition ”x is a philosopher”. If every element in x is a philosopher, then f(x) holds true for all values of x, expressed as; ∀x.f(x)
f(x) being true for at least one x is expressed as;

Note the symbolic or algebraic way of expressing the same things as in everyday language, but much with much shorter sentences and clearer meaning.
Although the two systems are not analogous, since everyday language is much open to ambiguities, as we shall see. In formal logic, a double negation, !(!p), is equal to an affirmation of p. While in ordinary language, one can say for instance ”this doesn’t mean that i don’t want it”, and mean something different from saying ”this means i want it”. Clearly there’s an ambiguity in saying the former, for it shows uncertainty of whether you want something or not. While the latter is definitively affirmative. (The semantics of everyday language does not seem to follow the law of the excluded middle.)

I think one of the aims of creating a language of formal logic was to resolve the misunderstandings that may arise from this and similar examples. And i wouldn’t be surprised if there was a strong psychological drive to do this, since it would make communication with other individuals much more straight forward, and accessible to a contemplative and introverted mind. (But that is not to say i know much about the enigmatic personality of Mr. Wittgenstein.)
It’s also worth mentioning that other interpretations exist of what formal/mathematical logic attempted to do. Douglas Hofstadter wrote in his famous Gödel, Escher, Bach: an Eternal Golden Braid (GEB) that mathematical logic ”began with the attempts to mechanize the thought processes of reasoning”. Thus not only communication would theoretically be influenced by this language, but also internal thoughts and the processes we use to arrive at conclusions and make decisions.
However, it’s important not to confuse formal logic, an a-priori system of deduction, with the psychological process of reasoning. They are of course in practice inevitably related, but separate as subjects of study. Furthermore, the assumption that all human thought follows logical steps is obviously idealistic, since the arguments our minds concoct are often prone to logical fallacies and cognitive biases. Although, going back to Wittgenstein, if one uses the more restrictive definition of a thought as ”a logical picture of facts”, it can be argued that all thoughts are logical.
But what if you think of something that doesn’t correspond to a fact about the world, such as an invisible pink unicorn, is that not a thought? Are you merely imagining things, which is different from the act of thinking? Or to complicate matters, you could be thinking of something you believe is a fact, but later turns out to not be the case. Does this thought spontaneously disappear from the category of thoughts in a puff of logic? Rather than having found a loophole in the TLP, these are probably cases Wittgenstein would have dismissed of as nonsensical, ”whereof one cannot speak, thereof one must be silent”.
Other such cases include the broad domains of ethics, metaphysics and aesthetics, which by their subjective nature, do not correspond to facts about the world. The following (Wittgensteinean) position not to speak of these topics, since they lie outside the limits of language, could be viewed as an ethical stance in itself. (Although this would put ethics in the ”nonsense” category which seems a bit absurd or at least nihilistic.)

I’m feeling a gradual loss of interest in finishing writing this, maybe because of indolence or since i had no planned structure to begin with. Also, i noticed that i have been alternating between writing about the system created in TLP, and within it. But i have been writing mostly about it, perhaps not as good a way to make sense of the contents.
Describing the content and terminology used in the book, is what i mean by being within the system. Here, one should not mix personal interpretation with the words of the original author, to convey as accurate a sample as possible of what it is like to read the book, and to delve into the thoughts of the author. When writing about the system, however, one can convey the personal experience of reading by describing the personal interpretation of the content, as opposed to the content itself. As well as relate it to other works or ideas.
This might be a blurry line though, probably more so than Wittgenstein’s line between sense and nonsense. Stepping outside the TLP, one does not simply transcend the limits of language (although it feels quite liberating to escape the state of confusion).

It was probably a bad idea to start with the TLP to get acquainted with formal logic, since it’s very purpose was to criticize philosophy and point out its limitations. That said, i don’t think it’s completely destroyed my curiosity, and any suggestions for modern textbooks on formal logic would be greatly appreciated in the comments section. Thanks for reading.

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